On the Method of Multiplier-enlargement and Approximation of Unbounded Continuous Functions

By combining the classical appropriate functions “1, x, x 2” with the method of multiplier enlargement, this paper establishes a theorem to approximate any unbounded continuous functions with modified positive linear operators. As an example, Hermite Fejér interpolation polynomial operators are analysed and studied, and a general conclusion is obtained.

Continuous Piecewise Linear Approximation of BV Function

Nonlinear approximation is widely used in signal processing. Real-life signals can be modeled as functions of bounded variation. Thus the variable knot of approximating function could be self- adaptively chosen by balancing the total variation of the target function. In this paper, we adopt continuous piecewise linear approximation instead of the existing piecewise constants approximation. The results of experiments show that this new method is superior to the old one.

On Approximation of Nonbounded Continuous Functions

This paper generalizes the basic principle of multiplier-enlargement approach to approximating any nonbounded continuous functions with positive linear operators, and as an example, Bernstein polynomial operators are analysed and studied. This paper gives a certain theorem as a general rule to approximate any nonbounded continuous functions.

APPROXIMATION OF CONTINUOUS FUNCTIONS BY GENERALIZED KARAMATA MEANS

A sufficient condition for the order of approximation of a continuous 2π periodic function with a given majorant for the modulus of continuity by the [F, d_n] means of its Fourier serier to be of Jackson order is obtained. This sufficient condition is shown to be not enough for the order of approximation by partial sums of their Fourier series to be of Jackson order. The error estimate is shown to be the best possible.

RATIONAL FUNCTIONS OF BEST UNIFORM APPROXIMATION AND HOLOMORPHIC CONTINUATION OF FUNCTIONS

Let f be a function, continuous and real valued on the segment △,△ (-∞,∞) and {Rn} be the sequence of the rational functions of best uniform approximation to fon △ of order (n,n). In the present work, the convergence of {Rn} in the complex plane is considered for the special caseswhen the poles (or the zeros, respectively) of {Rn} accumulate in the terms of weak convergence of measures to acompact set of zera capacity.As a consequence, sufficient conditions for the holomorphic and the meromorphic continuability of fare given.

The construction and approximation of feedforward neural network with hyperbolic tangent function

In this paper, we discuss some analytic properties of hyperbolic tangent function and estimate some approximation errors of neural network operators with the hyperbolic tan- gent activation function. Firstly, an equation of partitions of unity for the hyperbolic tangent function is given. Then, two kinds of quasi-interpolation type neural network operators are con- structed to approximate univariate and bivariate functions, respectively. Also, the errors of the approximation are estimated by means of the modulus of continuity of function. Moreover, for approximated functions with high order derivatives, the approximation errors of the constructed operators are estimated.

The Jackson Inequality for the Best L^2-Approximation of Functions on [0,1] with the Weight x

Let L^2（[0, 1], x） be the space of the real valued, measurable, square summable functions on [0, 1] with weight x, and let n be the subspace of L2（[0, 1], x） defined by a linear combination of Jo（μkX）, where Jo is the Bessel function of order 0 and {μk} is the strictly increasing sequence of all positive zeros of Jo. For f ∈ L^2（[0, 1], x）, let E（f, n） be the error of the best L2（[0, 1], x）, i.e., approximation of f by elements of n. The shift operator off at point x ∈[0, 1] with step t ∈[0, 1] is defined by T（t）f（x）=1/π∫0^π f（√x^2 ＋t^2-2xtcosO）dθ The differences （I- T（t））^r/2f = ∑j=0^∞（-1）^j（j^r/2）T^j（t）f of order r ∈ （0, ∞） and the L^2（[0, 1],x）- modulus of continuity ωr（f,τ） = sup{｜｜（I- T（t））^r/2f｜｜：0≤ t ≤τ] of order r are defined in the standard way, where T^0（t） = I is the identity operator. In this paper, we establish the sharp Jackson inequality between E（f, n） and ωr（f, τ） for some cases of r and τ. More precisely, we will find the smallest constant n（τ, r） which depends only on n, r, and % such that the inequality E（f, n）≤ n（τ, r）ωr（f, τ） is valid.

APPROXIMATION OF CONVEX TYPE FUNCTION BY PARTIAL SUMS OF FOURIER SERIES

The concept of convex type function is introduced in this paper,from which a kin d of convex decomposition approach is proposed.As one of applications of this a pproach,the approximation of the convex type function by the partial sum of its Fourier series is inves tigated.Moreover,the order of approximation is describe d with the 2th continuous modulus.

WEIGHTED APPROXIMATION OF RANDOM FUNCTIONS

Let (Ω, A, P) be a probability space, X(t, ω) a random function continuous in probability for t∈[0,+∞) or (-∞,+∞)(ω∈Ω), and F(t) a positive function continuous for t∈[0,+∞) or (-∞, +∞). If X(t, ω) and F(t) verify certain conditions, then there exists a sequence {Qn(t,ω)} of random polynomials such that we have almost surely: for t∈[0,+∞) or (-∞, +∞), lim|X(t, ω)-Qn(t, ω)|/F(t)=0.

Sub-Differential Characterizations of Non-Smooth Lower Semi-Continuous Pseudo-Convex Functions on Real Banach Spaces

In this paper, we characterize lower semi-continuous pseudo-convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the pseudo-monotonicity of its Clarke-Rockafellar Sub-differential. We extend the results on the characterizations of non-smooth convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the monotonicity of its sub-differentials to the lower semi-continuous pseudo-convex functions on real Banach spaces.

On Approximation by Reciprocals of Polynomials with Positive Coefficients

In order to study the approximation by reciprocals of polynomials with real coefficients, one always assumes that the approximated function has a fixed sign on the given interval. Sometimes, the approximated function is permitted to have finite sign changes, such as l（l ≥ 1） times. Zhou Songping has studied the case l=1 and l≥2 in L^p spaces in order of priority. In this paper, we studied the case l ≥2 in Orlicz spaces by using the function extend, modified Jackson kernel, Hardy-Littlewood maximal function, Cauchy-Schwarz inequality, and obtained the Jackson type estimation.

THE SHARP JACKSON INEQUALITY FOR L^2-APPROXIMATION ON THE PERIODIC CYLINDER

We consider Jackson inequality in L^2 （B^d×T, Wκ,μ^B （x））, where the weight function Wκ,μ^B （X） is defined on the ball B^d and related to reflection group, and obtain the sharp Jackson inequalityEn-1,m-1（f）2≤κn,m（τ,r）ωr（f,t）2,τ≥2τn,λ,where Tn,λ is the first positive zero of the Gegenbauer cosine polynomial Cn^λ （cos θ）（n ∈ N）.

Effects of Surface Charges on the Seawater Desalination Using Functionalized Graphene

In this paper, we investigate the selection rule for desalinating seawater using functionalized graphene sheet as a semi-permissible membrane. Both the applied mathematical modeling and MD simulations will be used to determine the acceptance conditions for water molecule or sodium ion permeating into the functionalized graphene. Both the Lennard-Jones potential and Coulomb forces are considered by taking into accounts the major molecular and ionic interactions between molecules, ions and functionalized graphene sheet. The continuous approximation will then be used to coarse grain most significant molecular and ionic interactions so that the multi-body problems could be simplified into several two-body problems and the 3D motions are reduced into degenerated 1D motion. Our mathematical model and simulations show that the negatively charged graphene always accepts sodium ions and water;however the permeability of water molecules and sodium ions becomes very sensitive to the presence of positive charges on the graphene.

改造的Catmull-Rom插值基函数及其逼近性质

此文利用多项式再生性和局部支撑性对著名的Catmull-Rom插值基函数进行改造,获得新的插值基函数,生成新的插值算子,并求得新插值算子精确的Lebesgue常数,估计了其逼近连续函数的收敛速度,理论和数值例子都表明新插值算子的逼近效果优于其他各类插值算子.

关于连续过程神经元网络的一些理论问题

针对输入/输出均为连续时间函数的非线性系统信号处理和建模问题,提出了一种连续过程神经元和过程神经元网络模型.连续过程神经元的输入/输出均为连续时间函数,其时空聚合运算能同时反映连续时变输入信号的空间聚合作用和输入过程中的时间累积效应,可实现输入/输出之间非线性实时或若干时间单元延迟的映射关系.文中给出了一种输入输出均为连续时间函数的前馈过程神经元网络模型,并证明了相应的连续性,函数逼近能力和计算能力等性质定理.

过程神经元网络的若干理论问题

文章提出一种过程神经元模型 ,其输入为与时间有关的函数或过程 ,它是传统人工神经元模型在时间域上的扩展。基于这种过程神经元模型 ,给出了一种仅含一个隐层的前馈型过程神经网络模型 ,即基展开过程神经元网络模型。证明了相应的连续性定理 ,逼近定理 ,计算能力定理等。

基于连分式分解的1/2~n阶模拟分抗逼近电路设计

应用连分式理论设计1/2n阶模拟分抗逼近电路。基于分数阶微积分理论,推导理想模拟分抗的网络函数。对1/2阶理想分抗的网络函数进行连分式分解,得到相应模拟分抗逼近电路的网络函数,并将其推广到1/2n阶。采用无源RC器件设计电路的具体结构,并通过multisim10仿真。实验结果证明,由连分式分解理论设计的1/2n阶分抗逼近电路具有良好的幅频响应和相频响应,能有效地逼近理想分抗。

基于核方法的连续动作Actor-Critic学习

强化学习算法通常要处理连续状态及连续动作空间问题以实现精确控制.就此文中结合Actor-Critic方法在处理连续动作空间的优点及核方法在处理连续状态空间的优势,提出一种基于核方法的连续动作Actor-Critic学习算法(KCACL).该算法中,Actor根据奖赏不作为原则更新动作概率,Critic采用基于核方法的在线选择时间差分算法学习状态值函数.对比实验验证该算法的有效性.

连续空间中的一种动作加权行动者评论家算法

经典的强化学习算法主要应用于离散状态动作空间中.在复杂的学习环境下,离散空间的强化学习方法不能很好地满足实际需求,而常用的连续空间的方法最优策略的震荡幅度较大.针对连续空间下具有区间约束的连续动作空间的最优控制问题,提出了一种动作加权的行动者评论家算法(Action Weight Policy Search Actor Critic,AW-PS-AC).AW-PS-AC算法以行动者评论家为基本框架,对最优状态值函数和最优策略使用线性函数逼近器进行近似,通过梯度下降方法对一组值函数参数和两组策略参数进行更新.对两组策略参数进行加权获得最优策略,并对获得的最优动作通过区间进行约束,以防止动作越界.为了进一步提高算法的收敛速度,设计了一种改进的时间差分算法,即采用值函数的时间差分误差来更新最优策略,并引入了策略资格迹调整策略参数.为了证明算法的收敛性,在指定的假设条件下对AW-PS-AC算法的收敛性进行了分析.为了验证AW-PS-AC算法的有效性,在平衡杆和水洼世界实验中对AW-PS-AC算法进行仿真.实验结果表明AW-PS-AC算法在两个实验中均能有效求解连续空间中近似最优策略问题,并且与经典的连续动作空间算法相比,该算法具有收敛速度快和稳定性高的优点.

关于Bernstein-Bézier算子对一类绝对连续函数的逼近

Bernstein-Bézier算子是一种重要的逼近算子,在计算机辅助几何设计中也扮演了重要角色.为了进一步了解它的理论及其逼近性质,研究了它对一类绝对连续函数的逼近.本文主要利用经典的Bojanic-Cheng分解方法,结合分析技术,分别讨论了Bernstein-Bézier算子在0<α≤1及α≥1时,对这类绝对连续函数的逼近.首先扩展了文献Liu的结果,得到了Bernstein-Bézier算子的一阶中心绝对矩B(nα)(t-x,x);接着估计了另外一项B(nα)(t∫xφx(u)du,x),最后得到了比较精确的收敛价.